5.2 - True/False

If lambda + 5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.

False - -5 is an eigenvalue

An elementary row operation on A does not change the determinant.

False - Interchanging rows and multiplying a row by a scalar changes the determinant. (Interchanging rows changes the sign and scalar multiplication multiplies the determinant by that same scalar.)

If A is 3 x 3, with columns a1, a2, and a3, then detA equals the volume of the parallelepiped determined by a1, a2, and a3.

False - It equals the ABSOLUTE VALUE of the determinant (since volume is always positive)

A row replacement operation on A does not change the eigenvalues.

False - Row replacements do not change the eigenvalues, since the eigenvalues are the roots of the characteristic polynomial formed by det(A - lambda* I)

A and B are n x n matrices. The determinant of A is the product of the diagonal entries in A.

False - This is only the case when A is a triangular matrix, or in reduced row echelon form.

det A^T = (-1)detA.

False - detA^T = detA

(detA)(detB) = detAB.

True - This is a property of determinants

The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.

True - by the definition